In this talk, we consider the Ising model at critical temperature with external magnetic field ha^{15/8} on the square lattice with lattice spacing a. We show that the truncated two-point function in this model decays exponentially with a rate independent of a. If time permits, we will discuss the problem of exponential decay in the near-critical scaling limit (as a approaches 0). This is joint work with Federico Camia and Chuck Newman.

We explain how the Edge-reinforced random walk, introduced by Coppersmith and Diaconis in 1986, is related to several models in statistical physics, namely the supersymmetric hyperbolic sigma model studied by Disertori, Spencer and Zirnbauer (2010), the random Schrödinger operator and Dynkin's isomorphism. These correspondences enable to show recurrence/transience results on the Edge-reinforced random walk, and also to provide insight into these models. This work is joint with Christophe Sabot, and part of it is also in collaboration with Margherita Disertori, Titus Lupu and Xiaolin Zeng.

In the first part of my talk I will focus on the latest developments in the study of Glauber dynamics of Stochastic Ising model on various graphs, explain its importance and review recent achievements. The second half of the talk will be dedicated to describe and discuss several important open problems, both on microscopic and macroscopic level, which are of importance to physics, mathematics, and theoretical computer science.

We introduce a measure on the unit circle that encodes the Beta-ensemble of every size. We will show connections to orthogonal polynomials, random walks, the Gaussian free field... In particular, we will present a result regarding the Hausdorff dimension of this measure, and introduce open questions.

The Once-reinforced random walk was introduced in the nineties as an a priori simplistic model of random walk with memory. Since then, only a few results have been proved, but very interesting features have been conjectured, some of them related to the geometry of the range.We will give an overview of the current knowledge on these questions, eventually focusing on a recent result of Vladas Sidoravicius and myself.

We consider a version of low temperature solid on solid 2+1 randominterface coupled with high and low density bulk Bernoulli fields, which is intendedto model flat faces of microscopic Wulff crystals in three dimensional Ising model.As the total number of particles in the bulk is tuned, the surface undergoes an infinitesequence of first order transitions, which correspond to spontaneous creation of macroscopicsize facets. The 1/3-scaling behaviour of ensembles of microscopic level lines along edgesof the confining box is also discussed.

Based on joint works with Senya Shlosman, Yvan Velenik and Vitali Wachtel.